Concept explainers
The compressed-air tank AB has a 250-rnm outside diameter and an 8-mm wall thickness. It is fitted with a collar by which a 40-kN force P is applied at B in the horizontal direction. Knowing that the gage pressure inside the tank is 5 MPa, determine the maximum normal stress and the maximum shearing stress at point K.
Fig. P7.124
Find the maximum normal stress and maximum shear stress at point K.
Answer to Problem 124P
The maximum normal stress and maximum shear stress at point K are
Explanation of Solution
Given information:
The outer diameter (d) of the tank is
The wall thickness (t) of the tank is
The magnitude of the load P is
The gage pressure (p) inside the tank is
Calculation:
Consider an element at point K.
Show the stress acting on the point K as shown in Figure 1.
Calculate the inner radius of the vessel
Substitute
Show the expression for hoop stress acting on the tank as shown below.
Substitute
Show the expression for longitudinal stress acting on the vessel as shown below.
Substitute
Calculate the stress due to bending at point K as follows:
The point K lies in the neutral axis, then
The stress due to bending at point K,
Calculate the stress due to transverse shear as follows:
Consider the shear force acting on the tank is denoted by V. Then,
Consider the distance between the centre of the circular cross-section to the inner and outer surface of the wall are denoted by
Calculate the shear flow Q using the relation:
Substitute
Calculate the moment of inertia I of the cross-section using the relation:
Substitute
Calculate the shear stress using the relation:
Substitute
Show the total hoop stress
Calculate the radius of the Mohr circle using the relation:
Substitute
Show the expression for average stress acting on the vessel as shown below.
Substitute
Consider the principal stress are denoted by
Calculate the value of the
Substitute
Calculate the value of the
Substitute
The stress at point C is 0. Then,
Compare the value of stress
Get the maximum and minimum value of the stress as follows:
Thus, the maximum normal stress in the tank is
Calculate the maximum shear stress in the tank using the relation:
Substitute
Thus, the maximum shear stress in the tank is
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Chapter 7 Solutions
Mechanics of Materials, 7th Edition
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